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Mirrors > Home > MPE Home > Th. List > cleljust | Structured version Visualization version GIF version |
Description: When the class variables in definition df-clel 2870 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2112 with the class variables in wcel 2111. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 2011 in order to remove dependencies on ax-10 2142, ax-12 2175, ax-13 2379. Note that there is no disjoint variable condition on 𝑥, 𝑦, that is, on the variables of the left-hand side, as should be the case for definitions. (Revised by BJ, 29-Dec-2020.) |
Ref | Expression |
---|---|
cleljust | ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ1 2118 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)) | |
2 | 1 | equsexvw 2011 | . 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦) |
3 | 2 | bicomi 227 | 1 ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: dfclel 2871 |
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